# Number Theory

## Explore the powers of divisibility, modular arithmetic, and infinity.

Back

#### Overview

This course starts at the very beginning — covering all of the essential tools and concepts in number theory, and then applying them to computational art, cryptography (code-breaking), challenging logic puzzles, understanding infinity, and more!

### Topics covered

• Divisibility Shortcuts
• Exploring Infinity
• Factor Trees
• Fermat's Little Theorem
• Greatest Common Divisor
• Least Common Multiple
• Modular Arithmetic
• Modular Congruence
• Modular Inverses
• Prime Factorization
• The 100 Doors Puzzle
• Totients

46

410+

1. 1

## Introduction

In these warmups, if you know the trick, you'll finish each problem in seconds!

1. ## Last Digits

Use shortcuts to find just the last digit of each answer – there's no need to calculate the rest!

1
2. ## Secret Messages

Look for patterns, and when you think you've found one, use it to decode the message!

2
3. ## Rainbow Cycles

Investigate the coloring rules that apply when you do math on a rainbow-striped number grid.

3
2. 2

## Factorization

Every integer greater than 1 has a unique name that can be written down in primes.

1. ## Divisibility Shortcuts (I)

How much can you learn about a number if you can only see its last digit?

4
2. ## Divisibility Shortcuts (II)

Review the divisibility shortcuts that apply when you're dividing by a power of 2 or 5.

5
3. ## Divisibility by 9 and 3

Explore the pattern of what remainders remain when you divide powers of 10 by 9 or 3.

6
4. ## Last Digits

Apply divisibility rules as well your own logic to determine just the last digit of each solution.

7
5. ## Arithmetic with Remainders

How do the remainders of an operation's inputs impact the remainder of the calculation output?

8
6. ## Digital Roots

Investigate surprising patterns that surface when you calculate digital roots.

9
7. ## Factor Trees

Factor each number one step at a time until every piece that you have is a prime.

10
8. ## Prime Factorization

Learn to use factorization as a versatile problem-solving tool.

11
9. ## Factoring Factorials

Since they're defined as products, what happens when you factor them?

12
10. ## Counting Divisors

Learn a quick technique for determining how many different divisors a number has.

13
11. ## 100 Doors

Imagine a long hallway with 100 closed doors numbered 1 to 100...

14
12. ## How Many Prime Numbers Are There?

Are there finitely many prime numbers or infinitely many of them, and how can you be sure either way?

15
3. 3

## GCD and LCM

Explore, apply, and relate the GCD and LCM.

1. ## 100 Doors Revisited

Again, imagine that long hallway of doors, but this time focus your attention on exactly who does what.

16
2. ## The LCM

Build intuition for where least common multiples appear in both abstract and real-life contexts.

17
3. ## Billiard Tables

Explore how the path of a ball bouncing around a pool table is affected by the table's dimensions.

18
4. ## The GCD

Use prime factorization as a tool for finding the greatest common divisors of pairs of numbers.

19
5. ## Dots on the Diagonal

When you draw a right triangle on a grid of dots, how many dots does does the hypotenuse cut through?

20
6. ## Number Jumping (I)

When do these jumping rules allow you to reach every number on the number line?

21
7. ## Number Jumping (II)

Develop a systematic procedure to determine the smallest positive integer that you can reach by jumps.

22
8. ## Number Jumping (III)

What's the pattern to these answers? What's going on in the big picture here?

23
9. ## Relating LCM and GCD

Understand how the GCD and LCM are related by thinking about factors arranged in a Venn diagram.

24
10. ## Billiard Tables Revisited (I)

Explore the patterns created when pool balls "paint" the squares they touch as they roll.

25
11. ## Billiard Tables Revisited (II)

How do you get back "home" to the bottom left pocket?

26
4. 4

## Modular Arithmetic (I)

The danger of cyclic systems: one step too far and you're back where you started!

1. ## Times and Dates

Time, as measured by a clock or calendar, is "modular," so let's start there...

27
2. ## Modular Congruence

What happens when you wrap an infinite number line around a one-unit square?

28
3. ## Modular Arithmetic

Learn and practice doing arithmetic in the modular world.

29
4. ## Divisibility by 11

Review the rules for arithmetic with remainders and uncover the peculiar divisibility rule for 11.

30
5. ## Star Drawing (I)

You probably know how to draw a 5-pointed star, but what about an 8 or 12 or 30-pointed star?

31
6. ## Star Drawing (II)

Learn a general formula for the number of points a star will have.

32
7. ## Star Drawing (III)

How many different 60-point stars can you make, drawing just one path on a circle of 60 points?

33
8. ## Die Hard Decanting (I)

The challenge is to measure out a specific quantity of liquid using only a few types of legal moves.

34
9. ## Die Hard Decanting (II)

Which pairs of containers can measure out any quantity of liquid and which ones have limited use?

35
5. 5

## Modular Arithmetic (II)

Considering the remainder "modulo" an integer is a powerful tool with many applications!

Explore a concept that's lurking beneath the surface of both star drawing and decanting puzzles.

36
2. ## Modular Multiplicative Inverses

Can normal equations with no integer solutions be converted into congruences that DO have solutions?

37
3. ## Multiplicative Cycles

What does exponentiation look like in modular arithmetic?

38
4. ## Fermat's Little Theorem

Use the factorization of a number to determine how many small numbers are relatively prime to it.

39
5. ## Totients

This isn't Fermat's fearsome Last Theorem, but it still packs a big punch for a little guy!

40
6. ## Last Digits Revisited

Use Euler's theorem to quickly find just the last few digits of enormous exponential towers!

41
7. ## Perfect Shuffling

Leverage what you know about totients to find any card in the deck after a series of perfect shuffles.

42
6. 6

## Exploring Infinity

Explore one of the most misunderstood concepts in math - infinity.

1. ## Counting to Infinity

To understand cardinal infinity, first start by counting and comparing finite sets.

43
2. ## Multiple Infinities

Explore a crazy world of numbers that contains infinitely many infinities, both small and large.

44
3. ## Hilbert's Hotel

Help Hilbert use his hotel that has infinitely many rooms to host infinitely many sleepy guests.

45
4. ## Infinitely Large

Can a betting game have an infinite expected value?

46
7. 7